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1. CJM 2007 (vol 59 pp. 85)
On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.
Keywords:Series, convergence, almost alternating, convex, continued fractions Categories:40A05, 11A55, 11B83 |
2. CJM 2002 (vol 54 pp. 1305)
Continued Fractions Associated with $\SL_3 (\mathbf{Z})$ and Units in Complex Cubic Fields Continued fractions associated with $\GL_3 (\mathbf{Z})$ are
introduced and applied to find fundamental units in a two-parameter
family of complex cubic fields.
Keywords:fundamental units, continued fractions, diophantine approximation, symmetric space Categories:11R27, 11J70, 11J13 |